Finite Group Databasebeta

← #27826

Subgroups — #27826

Order 256.

353 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2563C211yesmaximal, characteristic
2#2550C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2101C211yesmaximal, characteristic, complemented
5#2025C211yesmaximal, characteristic
6#2023C211yesmaximal, characteristic
7#1993C211yesmaximal, characteristic, complemented
8#1993C211yesmaximal, characteristic, complemented
9#1991C211yesmaximal, characteristic, complemented
10#1988C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic
12#1934C211yesmaximal, characteristic
13#1804C211yesmaximal, characteristic
14#1771C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×#102C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#97C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#398—21no—
29#395—21no—
30#393—21no—
31#393—21no—
32#391—21no—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#389—21no—
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#382C2211yescharacteristic
39#382C2211yescharacteristic
40#382C2211yescharacteristic
41#380—21no—
42#380—21no—
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71#125—22no—
72#125—41no—
73C4×D4—22no—
74C4×D4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C4⋊C8—22no—
100#102—22no—
101#102—41no—
102#102—41no—
103#102Q814yescomplemented
104#97—22no—
105#97—41no—
106C4×C8—41no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95D411yescharacteristic
117#95D411yescharacteristic
118C24—21no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—81no—
123C2×Q8—81no—
124C2×Q8—81no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4C2×Q812yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2411yescharacteristic
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8C2×Q812yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4C2×Q812yes—
181C4⋊C4#4112yes—
182C22⋊C4—42no—
183C22⋊C4—81no—
184C22⋊C4—81no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—81no—
192C42#4112yes—
193C42#4112yes—
194C23—22no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23C22×D411yescharacteristic
202Q8—84no—
203Q8—84no—
204D4—22no—
205D4—22no—
206D4—41no—
207D4—41no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—84no—
246C2×C4#12412yes—
247C2×C4#12412yes—
248C2×C4C22×Q811yescharacteristic
249C2×C4C2×#4111yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C4H(5,𝔽2)11yescharacteristic
253C2×C42−1+411yescharacteristic
254C8—22no—
255C8—41no—
256C8—41no—
257C8—41no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22C2×#12411yescharacteristic
266C22#54611yescharacteristic
267C22D4×Q811yescharacteristic
268C22#57511yescharacteristic
269C22#57511yescharacteristic
270C22#57611yescharacteristic
271C22#57711yescharacteristic
272C4—22no—
273C4—22no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4#53311yescharacteristic
286C4#54211yescharacteristic
287C4#55411yescharacteristic
288C4#55611yescharacteristic
289C2—82no—
290C2#249211yescharacteristic
291C2#313611yescharacteristic
292C2#314011yescharacteristic
293C2#331911yescharacteristic
294C2#332111yescharacteristic
295C2#338811yescharacteristic
296C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2550C2semidirect
3. #2101C2semidirect
4. #2101C2semidirect
7. #1993C2semidirect
8. #1993C2semidirect
9. #1991C2semidirect
10. #1988C2semidirect
14. #1771C2semidirect
103. #102Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#25632
#25502
#21012
#20252
#20232
#19932
#19912
#19882
#19362
#19342
#18042
#17712
#17622